Given that

\begin{align*}
\frac{1}{x}+\frac{1}{y}&=3,\\
xy+x+y&=4,
\end{align*}

compute $x^2y+xy^2$.
The first equation becomes

$$\frac{x+y}{xy}=3\Rightarrow x+y=3xy$$

Substituting into the second equation,

$$4xy=4\Rightarrow xy=1$$

Thus $x+y=3$.

The quantity we desire factors as $xy(x+y)$, so it is equal to $1(3)=\boxed{3}$.